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How is a negative integer power on a base related to the corresponding positive integer power on that base?

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Scientific notation is used to express numbers that are really really big. It can also be used to express numbers that are really really small. 

An example is the speed of light.

Light travels at a really fast speed, at 3.0 x 108 meters/second.

When you write this out, it becomes 300000000 meters/second.

Scientific notations are used to simplify these types of numbers into numbers we can easily recognize (without having to count the number of zeros).

Another example is the size of an electron, which is theoretically 2.82 x 10-15m.

The negative integer power signifies the number of times the decimal has to move to the left in order for the number to be written in the long form. 

When you write the 2.82 x 10-15m out, it becomes 0.00000000000000282 m.

See how hard it is to count the number of zeros?

That is why these types of numbers are written in the form called scientific notation.